TITLE: Is the adjoint L-function on GL(m) holomorphic?
QUESTION [14 upvotes]: Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of degree $m^2-1$.
Is $L(s,\pi,\mathrm{Ad})$ holomorphic on the entire complex plane? As far as I know, this is known for $m=2$ by the work Shimura and Gelbart-Jacquet. What is know on the case of $m>2$?

REPLY [9 votes]: I think this is widely open. Flicker has a conditional result under certain cases of the Artin conjecture for Artin $L$-functions, see the Theorem on Page 232 of Pacific J. Math. 154 (1992). In particular, his result shows that the adjoint $L$-function is entire for $m=2,3,4$.
Added. As GFS remarked, the work of Ginzburg et al. shows, by the method of integral representations, that the adjoint $L$-function is entire for $m=3,4,5$.